Is there any way to get across to the other side of the shaft?

Hint 1 of 9:

Yes, there is.

Hint 2 of 9:

Look closely at the doors on the side you're currently on.

Hint 3 of 9:

Or, more precisely, look just below the doors and to the left.

Hint 4 of 9:

You'll see some controls there.

Hint 5 of 9:

Try turning the wheel on the controls.  (You'll want to do this on the third
level -- this doesn't work anywhere else, as all of the other controls are
disabled.)

Hint 6 of 9:

It looks like the wheel is stuck.

Hint 7 of 9:

You may want to loosen it.

Hint 8 of 9:

Oil might work here.

Hint 9 of 9:

The gourds in the theatre on the third level are filled with oil.